


from __future__ import annotations

from collections.abc import Callable

lowerCAmelCase__ 						=					list[list[float | int]]
def 				_UpperCAmelCase							(UpperCamelCase__	:    Union[str, Any]							,				UpperCamelCase__	:    int		):
       _A :			Dict							       =					len(UpperCamelCase__		)
       _A :			Optional[Any]							       =					[[0 for _ in range(size + 1		)] for _ in range(UpperCamelCase__		)]
       _A :			Union[str, Any]							       =					42
       _A :			int							       =					42
       _A :			Tuple							       =					42
       _A :			int							       =					42
       _A :			str							       =					42
       _A :			Optional[int]							       =					42

       for row in range(UpperCamelCase__		):
              for col in range(UpperCamelCase__		):
                     _A :			List[str]							       =					matrix[row][col]

              _A :			str							       =					vector[row][0]

       _A :			str							       =					0
       _A :			Union[str, Any]							       =					0
       while row < size and col < size:
              # pivoting
              _A :			Optional[int]							       =					max((abs(augmented[rowa][col]		), rowa) for rowa in range(UpperCamelCase__							,				UpperCamelCase__		)		)[
                  1
              ]
              if augmented[pivot_row][col] == 0:
                     col += 1
                     continue
              else:
                     _A       ,     _A :			List[str]							       =					augmented[pivot_row], augmented[row]

              for rowa in range(row + 1							,				UpperCamelCase__		):
                     _A :			Optional[Any]							       =					augmented[rowa][col] / augmented[row][col]
                     _A :			Dict							       =					0
                     for cola in range(col + 1							,				size + 1		):
                            augmented[rowa][cola] -= augmented[row][cola] * ratio

              row += 1
              col += 1

       # back substitution
       for col in range(1							,				UpperCamelCase__		):
              for row in range(UpperCamelCase__		):
                     _A :			int							       =					augmented[row][col] / augmented[col][col]
                     for cola in range(UpperCamelCase__							,				size + 1		):
                            augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
       return [
           [round(augmented[row][size] / augmented[row][row]							,				10		)] for row in range(UpperCamelCase__		)
       ]
def 				_UpperCAmelCase							(UpperCamelCase__	:    List[str]		):
       _A :			List[Any]							       =					len(UpperCamelCase__		)
       _A :			Optional[Any]							       =					[[0 for _ in range(UpperCamelCase__		)] for _ in range(UpperCamelCase__		)]
       _A :			List[str]							       =					[[0] for _ in range(UpperCamelCase__		)]
       _A :			Dict							       =					42
       _A :			Tuple							       =					42
       _A :			str							       =					42
       _A :			Optional[int]							       =					42

       for x_val, y_val in enumerate(UpperCamelCase__		):
              for col in range(UpperCamelCase__		):
                     _A :			int							       =					(x_val + 1) ** (size - col - 1)
              _A :			Dict							       =					y_val

       _A :			List[str]							       =					solve(UpperCamelCase__							,				UpperCamelCase__		)

       def interpolated_func(UpperCamelCase__	:    List[str]		) -> int:
              return sum(
                  round(coeffs[x_val][0]		) * (var ** (size - x_val - 1))
                  for x_val in range(UpperCamelCase__		)		)

       return interpolated_func
def 				_UpperCAmelCase							(UpperCamelCase__	:    List[str]		):
       return (
           1
           - variable
           + variable**2
           - variable**3
           + variable**4
           - variable**5
           + variable**6
           - variable**7
           + variable**8
           - variable**9
           + variable**10
       )
def 				_UpperCAmelCase							(UpperCamelCase__	:    Optional[Any] = question_function							,				UpperCamelCase__	:    Any = 10		):
       _A :			List[str]							       =					[func(UpperCamelCase__		) for x_val in range(1							,				order + 1		)]

       _A :			Dict							       =					[
           interpolate(data_points[:max_coeff]		) for max_coeff in range(1							,				order + 1		)
       ]

       _A :			Tuple							       =					0
       _A :			Dict							       =					42
       _A :			List[str]							       =					42

       for poly in polynomials:
              _A :			List[Any]							       =					1
              while func(UpperCamelCase__		) == poly(UpperCamelCase__		):
                     x_val += 1

              ret += poly(UpperCamelCase__		)

       return ret


if __name__ == "__main__":
    print(f"{solution() = }")

'''simple docstring'''




import numpy as np






def __lowerCAmelCase					(				UpperCamelCase__							,    UpperCamelCase__							,    UpperCamelCase__							,    UpperCamelCase__							,    UpperCamelCase__							)      ->       str:
    __lowerCamelCase            =    int(np.ceil((x_end - xa) / h							)							)
    __lowerCamelCase            =    np.zeros((n + 1,)							)
    __lowerCamelCase            =    ya
    __lowerCamelCase            =    xa

    for k in range(UpperCamelCase__							):
        __lowerCamelCase            =    f(UpperCamelCase__							,    y[k]							)
        __lowerCamelCase            =    f(x + 0.5 * h							,    y[k] + 0.5 * h * ka							)
        __lowerCamelCase            =    f(x + 0.5 * h							,    y[k] + 0.5 * h * ka							)
        __lowerCamelCase            =    f(x + h							,    y[k] + h * ka							)
        __lowerCamelCase            =    y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
        x += h

    return y


if __name__ == "__main__":
  import doctest

  doctest.testmod()
